Electronic Communications in Probability

Local explosion in self-similar growth-fragmentation processes

Jean Bertoin and Robin Stephenson

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Markovian growth-fragmentation processes describe a family of particles which can grow larger or smaller with time, and occasionally split in a conservative manner. They were introduced in [3], where special attention was given to the self-similar case. A Malthusian condition was notably given under which the process does not locally explode, in the sense that for all times, the masses of all the particles can be listed in non-increasing order. Our main result in this work states the converse: when this condition is not verified, then the growth-fragmentation process explodes almost surely. Our proof involves using the additive martingale to bias the probability measure and obtain a spine decomposition of the process, as well as properties of self-similar Markov processes.

Article information

Electron. Commun. Probab., Volume 21 (2016), paper no. 66, 12 pp.

Received: 19 February 2016
Accepted: 1 August 2016
First available in Project Euclid: 14 September 2016

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Zentralblatt MATH identifier

Primary: 60F17: Functional limit theorems; invariance principles 60G51: Processes with independent increments; Lévy processes 60J25: Continuous-time Markov processes on general state spaces 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

growth-fragmentation self-similarity branching process spine decomposition

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Bertoin, Jean; Stephenson, Robin. Local explosion in self-similar growth-fragmentation processes. Electron. Commun. Probab. 21 (2016), paper no. 66, 12 pp. doi:10.1214/16-ECP13. https://projecteuclid.org/euclid.ecp/1473854582

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